Finite additivity of a set function ($F(A\cup B)=F(A)+F(B)(A\cap B=\varnothing )$)

A real-valued set function on an algebra of sets is finitely additive if it has the displayed property for disjoint pairs. It follows that $F(\varnothing )=0$. On a finite ambient set, induction gives $F(A)={\sum }_{x\in A}F(\{x\})$, so arbitrary signed singleton weights describe all such functions. Summing the pointwise indicator function identity for the inclusion-exclusion principle against these weights proves the same formula for $F$.